Advertisements
Advertisements
प्रश्न
Find expected value and variance of X, the number on the uppermost face of a fair die.
उत्तर
Let X denote the number on uppermost face.
∴ Possible values of X are 1, 2, 3, 4, 5, 6.
Each outcome is equiprobable.
∴ P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = `(1)/(6)`
∴ Expected value of X
= E(X)
= \[\sum\limits_{i=1}^{6} x_i.\text{P}(x_i)\]
= `1 xx (1)/(6) + 2 xx (1)/(6) + 3 xx (1)/(6) + 4 xx (1)/() + 5 xx (1)/(6) + 6 xx (1)/(6)`
= `(1)/(6)(1 + 2 + 3 + 4 + 5 + 6)`
= `(21)/(6)`
= `(7)/(2)`
E(X2) = \[\sum\limits_{i=1}^{6} x_i^2.\text{P}(x_i)\]
= `1^2 xx (1)/(6) + 2^2 xx (1)/(6) + 3^2 xx (1)/(6) + 4^2 xx (1)/(6) + 5^2 xx (1)/(6) + 6^2 xx (1)/(6)`
= `(1)/(6)(1^2 + 2^ + 3^2 + 4^2 + 5^2 + 6^2)`
= `((6 xx 7 xx 13))/(6 xx 6)`
= `(91)/(6)`
∴ Variance of X
= Var(X)
= E(X2) – [E(X)]2
= `(91)/(6) - (7/2)^2`
= `(35)/(12)`.
APPEARS IN
संबंधित प्रश्न
State if the following is not the probability mass function of a random variable. Give reasons for your answer
| Z | 3 | 2 | 1 | 0 | −1 |
| P(Z) | 0.3 | 0.2 | 0.4 | 0 | 0.05 |
Find the mean number of heads in three tosses of a fair coin.
It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by
f (x) = `x^2 /3` , for –1 < x < 2 and = 0 otherwise
Verify whether f (x) is p.d.f. of r.v. X.
It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by
f (x) = `x^2/3` , for –1 < x < 2 and = 0 otherwise
Find probability that X is negative
Choose the correct option from the given alternative:
Find expected value of and variance of X for the following p.m.f.
| X | -2 | -1 | 0 | 1 | 2 |
| P(x) | 0.3 | 0.3 | 0.1 | 0.05 | 0.25 |
Solve the following problem :
A fair coin is tossed 4 times. Let X denote the number of heads obtained. Identify the probability distribution of X and state the formula for p. m. f. of X.
The following is the c.d.f. of r.v. X
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| F(X) | 0.1 | 0.3 | 0.5 | 0.65 | 0.75 | 0.85 | 0.9 |
*1 |
P (–1 ≤ X ≤ 2)
X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = _______
Fill in the blank :
If X is discrete random variable takes the value x1, x2, x3,…, xn then \[\sum\limits_{i=1}^{n}\text{P}(x_i)\] = _______
Fill in the blank :
E(x) is considered to be _______ of the probability distribution of x.
If r.v. X assumes values 1, 2, 3, ……. n with equal probabilities then E(X) = `("n" + 1)/(2)`
Solve the following problem :
The probability distribution of a discrete r.v. X is as follows.
| X | 1 | 2 | 3 | 4 | 5 | 6 |
| (X = x) | k | 2k | 3k | 4k | 5k | 6k |
Determine the value of k.
Solve the following problem :
Find the expected value and variance of the r. v. X if its probability distribution is as follows.
| x | – 1 | 0 | 1 |
| P(X = x) | `(1)/(5)` | `(2)/(5)` | `(2)/(5)` |
Solve the following problem :
Let the p. m. f. of the r. v. X be
`"P"(x) = {((3 - x)/(10)", ","for" x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).
Solve the following problem :
Let X∼B(n,p) If E(X) = 5 and Var(X) = 2.5, find n and p.
If a d.r.v. X takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5–x, where k is a constant, then P(X = 0) = ______
If the p.m.f. of a d.r.v. X is P(X = x) = `{{:(x/("n"("n" + 1))",", "for" x = 1"," 2"," 3"," .... "," "n"),(0",", "otherwise"):}`, then E(X) = ______
The following function represents the p.d.f of a.r.v. X
f(x) = `{{:((kx;, "for" 0 < x < 2, "then the value of K is ")),((0;, "otherwise")):}` ______
The p.m.f. of a random variable X is as follows:
P (X = 0) = 5k2, P(X = 1) = 1 – 4k, P(X = 2) = 1 – 2k and P(X = x) = 0 for any other value of X. Find k.
